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A Baer-Krull Theorem for Quasi-Ordered Groups

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Salma Kuhlmann
Salma Kuhlmann
Salma Kuhlmann
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Gabriel Lehéricy
Gabriel Lehéricy
Gabriel Lehéricy
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Abstract

We give group analogs of two important theorems of real algebra concerning convex valuations, one of which is the Baer-Krull theorem. We do this by using quasi-orders, which gives a uniform approach to valued and ordered groups. We also recover the classical Baer-Krull theorem from its group analog.
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Order (2018) 35:283–291
A Baer-Krull Theorem for Quasi-Ordered Groups
Salma Kuhlmann1·Gabriel Leh´
ericy1
Received: 21 March 2017 / Accepted: 30 May 2017 / Published online: 6 June 2017
© Springer Science+Business Media Dordrecht 2017
Abstract We give group analogs of two important theorems of real algebra concerning
convex valuations, one of which is the Baer-Krull theorem. We do this by using quasi-
orders, which gives a uniform approach to valued and ordered groups. We also recover the
classical Baer-Krull theorem from its group analog.
Keywords Valuations ·Ordered groups ·Quasi-orders ·Baer-Krull theorem
1 Introduction
The theories of field ordering and field valuation present some strong similarities but are
classically treated as separate subjects. However, in [3], Fakhruddin found a way of unifying
these two theories by using quasi-orders. He defined a quasi-ordered field as a field K
endowed with a total quasi-order (q.o) satisfying the following axioms:
(Q1)∀x(x ∼0⇒x=0)
(Q2)∀x,y,z(x yz⇒x+zy+z)
(Q3)∀x,y,z(x y∧0z) ⇒xz yz
He then showed that valued and ordered fields are particular instances of q.o fields, and
even showed that they are the only ones, so that the theory of q.o fields is very convenient
to unify the theories of valued fields with the theory of ordered fields. Two different notions
of quasi-ordered groups have been studied in [7]and[8], both of which can be seen as a
Gabriel Leh´
ericy
gabriel.lehericy@uni-konstanz.de
1Universitat Konstanz, Konstanz, Germany
DOI 10.1007/s11083-017-9432-5
Content courtesy of Springer Nature, terms of use apply. Rights reserved.
... It remains to show that H can be endowed with an order extending the one on G, so that ι is actually an embedding of asymptotic couples. For this, we use results from [13]. Since ψ is H-type, it is a coarsening of v G , so it follows from [13,Theorem 3.2] that ≤ induces an order on each B λ . ...
... For this, we use results from [13]. Since ψ is H-type, it is a coarsening of v G , so it follows from [13,Theorem 3.2] that ≤ induces an order on each B λ . By [13,Theorem 4.6], we can lift this family of orders (≤ λ ) λ∈ to H . ...
... Since ψ is H-type, it is a coarsening of v G , so it follows from [13,Theorem 3.2] that ≤ induces an order on each B λ . By [13,Theorem 4.6], we can lift this family of orders (≤ λ ) λ∈ to H . This gives us a group order on H whose restriction to ι( G) is exactly the order of G. Now if we restrict ι to G, we have an embedding of asymptotic couples ι : (G, ψ) → (H , ψ). ...
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We develop a notion of (principal) differential rank for differential-valued fields, in analog of the exponential rank developed in Kuhlmann (The Fields Institute Monograph Series 12, 2000) and of the difference rank developed in Kuhlmann (Groups, Modules and Model Theory—Surveys and Recent Developments in Memory of Rdiger Gbel, pp 399–414, 2017). We give several characterizations of this rank. We then give a method to define a derivation on a field of generalized power series and use this method to show that any totally ordered set can be realized as the principal differential rank of a H-field.
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... It remains to show that H can be endowed with an order extending the one on G, so that ι is actually an embedding of asymptotic couples. For this, we use results from [13]. Since ψ is H-type, it is a coarsening of v G , so it follows from [13, Theorem 3.2] that ≤ induces an order on each B λ . ...
... Since ψ is H-type, it is a coarsening of v G , so it follows from [13, Theorem 3.2] that ≤ induces an order on each B λ . By [13,Theorem 4.6], we can lift this family of orders (≤ λ ) λ∈Ψ to H. This gives us a group order on H whose restriction to ι( G) is exactly the order of G. Now if we restrict ι to G, we have an embedding of asymptotic couples ι : (G, ψ) → (H, ψ). ...
The differential rank of a differential-valued field
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  • Jul 2017
We develop a notion of (principal) differential rank for differential-valued fields, in analog of the exponential rank and of the difference rank. We give several characterizations of this rank. We then give a method to define a derivation on a field of generalized power series and use this method to show that any totally ordered set can be realized as the principal differential rank of a H-field.
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Compatibility of Quasi-Orderings and Valuations: A Baer-Krull Theorem for Quasi-Ordered Rings
Article
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  • ORDER
In a previous paper, we introduced the class of quasi-ordered commutative rings and proved that each such ring (R,⪯)(R,\preceq) is either an ordered ring or a valued ring. Here we take a further step in our investigation of this class. We develop the notion of ⪯\preceq-compatible valuations, leading to a definition of the rank of (R,⪯).(R,\preceq). We exploit it to establish a Baer-Krull Theorem; more precisely, fixing a valuation v on R, we describe all v-compatible quasi-orders on R. In case where the quasi-order is an order, this yields a generalization of the classical Baer-Krull Theorem for ordered fields. Else, if we restrict attention to quasi-orders that come from valuations, our results give rise to a complete characterization of all the coarsenings, respectively all the refinements, of a given valuation v on $R.
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